1. Field of the Invention
The present invention relates in general to the field of signal processing, and more specifically to a system and method for quantizing input signals using a delta sigma modulator that includes a non-monotonic quantizer.
2. Description of the Related Art
Many signal processing systems include delta sigma modulators to quantize an input signal into one or more bits. Delta sigma modulators trade-off increased noise in the form of quantization error in exchange for high sample rates and noise shaping. “Delta-sigma modulators” are also commonly referred to using other interchangeable terms such as “sigma-delta modulators”, “delta-sigma converters”, “sigma delta converters”, and “noise shapers”.
FIG. 1 depicts a conventional delta sigma modulator 100 that includes a monotonic quantizer 102 for quantizing a digital input signal x(n), where “x(n)” represents the nth input signal sample. The delta sigma modulator 100 also includes an exemplary fourth (4th) order noise shaping loop filter 104 that pushes noise out the signal band of interest. For audio signals, the signal band of interest is approximately 0 Hz to 20 kHz. The four feedback coefficients c0, c1, c2, and c3 set the poles of both the noise transfer function (NTF) and the signal transfer function (STF) of filter 104. In an audio application, the STF is a low-pass and generally all pole function. The NTF of filter 104 has four (4) zeros at DC (0 Hz). Typical high performance delta sigma modulators include fourth (4th) order and higher loop filters although filter 104 can be any order. The NTF distributes zeros across the signal band of interest to improve the noise performance of the delta sigma modulator 100.
FIG. 2 depicts the quantizer 102 modeled as a gain, g, multiplying the quantizer input signal s(n) plus additive white noise n. The quantizer output noise is then modeled as n/(1+z−1*g*H(z)). However, the quantizer output noise model often breaks down because the gain g is actually dependent upon the level (magnitude) of the input signal x(n). For low level input signals x(n), a tendency exists for the feedback signal from the quantizer 102 to be low, effectively making the gain high or breaking down the quantizer output noise model altogether. Because one-bit quantizers have no defined gain, a high gain for low level quantizer input signals is particularly bad because it can decrease the signal-to-noise ratio (SNR) of the delta sigma modulator 100.
Referring to FIGS. 1, 2, and 3, the quantizer 102 quantizes an input signal x(n) monotonically by making a decision to select the closest feedback value to approximate the input signal. In a one-bit delta sigma modulator, the quantizer has only two legal outputs, referred to as −1 and +1. Therefore, in a one-bit embodiment, quantizer 102 quantizes all positive input signals as a +1 and quantizes all negative input signals as −1. The quantization level changeover threshold 304 is set at DC, i.e. 0 Hz, and may be quantized as +1 or −1.
FIG. 3 graphically depicts a monotonic, two-level quantization transfer function 300, which represents the possible selections of each quantizer output signal y(n) from each quantizer input signal s(n). The diagonal line 302 depicts a monotonic unity gain function and represents the lowest noise quantization transfer function. “Monotonic” is defined by a function that, as signal levels increase, consists of either increasing quantizer output state transitions (“transitions”) or decreasing transitions, but not both increasing and decreasing transitions. To mathematically define “monotonically increasing” in terms of quantization, if the transfer function of the quantizer 102 is denoted as Q(s), then Q(s1)≧Q(s2), for all s1>s2, where “s1” and “s2” represent quantizer input signals. Mathematically defining “monotonically decreasing” in terms of quantization, if the transfer function of the quantizer 102 is denoted as Q(s), then Q(s1)≧Q(s2), for all s1<s2. Thus, in general, a monotonic quantization transfer function must adhere to Equation 1:Q(s1)≧Q(s2), for all |s1|>|s2|.  [Equation 1]
In many cases, dithering technology intentionally adds noise to the quantizer input signal s(n) to dither the output decision of quantizer 102. Adding dithering noise can help reduce the production of tones in the output signal y(n) at the cost of adding some additional noise to the delta sigma modulator loop because the quantization noise is generally increased. However, adding dithering noise to the quantizer does not convert a monotonic quantization transfer function into a non-monotonic quantization transfer function. Adding dithering noise merely changes the probability of some quantizer decisions. An alternative perspective regarding dither is to simply add a signal prior to quantization, which has no effect on the quantization transfer function.
Magrath and Sandler in A Sigma-Delta Modulator Topology with High Linearity, 1997 IEEE International Symposium on Circuits and Systems, Jun. 9–12, 1987 Hong Kong, (referred to as “Magrath and Sandler”) describes a sigma-delta modulator function that achieves high linearity by modifying the transfer function of the quantizer loop to include bit-flipping for small signal inputs to the quantizer. Magrath and Sandler discusses the compromise of linearity of the sigma-delta modulation process by the occurrence of idle tones, which are strongly related to repeating patterns at the modulator output and associated limit cycles in the system state-space. Magrath and Sandler indicates that injection of a dither source before the quantizer is a common approach to linearise the modulator. Magrath and Sandler discusses a technique to emulate dither by approximately mapping the dither onto an equivalent bit-flipping operation.
FIG. 4 graphically depicts the single non-monotonic region quantization transfer function 400 that emulates dither as described by Magrath and Sandler. Quantizer function 400 is necessarily centered around s(n)=0, as described by Magrath and Sandler, to emulate conventional dither. According to Magrath and Sandler, if the absolute value of the input (“|s(n)|” in FIG. 1) to the quantizer is less than B, a system constant, then the quantizer state is inverted as depicted by quantizer function 400.
Input signals s(n) to the quantizer 102 can be represented by probability density functions (PDFs). FIG. 5A depicts PDFs of each quantizer input signal s(n) during operation at small and large input signal levels. PDF 502 represents small signal levels for each signal s(n). The narrow PDF 502 can indicate high delta sigma modulator loop gain g. As the magnitude of signal levels for signal s(n) increase, the PDF of each signal s(n) changes from the narrow PDF 502 to the wider PDF 504.
FIG. 5B depicts a near ideal PDF 500 for each quantizer input signal s(n) because all signals are clustered around the quantization levels +1 and −1. Accordingly, the quantization noise n (error) is very small.